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What is the Product Rule for Derivatives: A Simple Explanation
The product rule is a fundamental concept in calculus that allows us to find the derivative of a product of two functions. It is an essential tool for solving problems involving rates of change and optimization. In this article, we will provide a simple explanation of the product rule for derivatives.
To understand the product rule, let’s start by considering two functions, f(x) and g(x), and their product h(x) = f(x) * g(x). The derivative of h(x) represents the rate of change of the product with respect to x.
The product rule states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function. Mathematically, it can be expressed as:
d/dx [f(x) * g(x)] = f'(x) * g(x) + f(x) * g'(x)
Here, f'(x) represents the derivative of f(x) with respect to x, and g'(x) represents the derivative of g(x) with respect to x.
To apply the product rule, follow these steps:
1. Identify the two functions that are being multiplied together.
2. Differentiate the first function with respect to x to find its derivative, f'(x).
3. Differentiate the second function with respect to x to find its derivative, g'(x).
4. Multiply the derivative of the first function, f'(x), by the second function, g(x).
5. Multiply the first function, f(x), by the derivative of the second function, g'(x).
6. Add the results from steps 4 and 5 to find the derivative of the product.
Let’s consider an example to illustrate the product rule. Suppose we have the functions f(x) = x^2 and g(x) = sin(x). We want to find the derivative of their product, h(x) = f(x) * g(x).
Step 1: Identify the functions: f(x) = x^2 and g(x) = sin(x).
Step 2: Differentiate the first function: f'(x) = 2x.
Step 3: Differentiate the second function: g'(x) = cos(x).
Step 4: Multiply f'(x) by g(x): 2x * sin(x).
Step 5: Multiply f(x) by g'(x): x^2 * cos(x).
Step 6: Add the results from steps 4 and 5: h'(x) = 2x * sin(x) + x^2 * cos(x).
So, the derivative of the product f(x) * g(x) is h'(x) = 2x * sin(x) + x^2 * cos(x).
The product rule is a powerful tool that allows us to find the derivative of a product of functions. It is particularly useful when dealing with complex functions or when solving optimization problems. By following the steps outlined above, you can easily apply the product rule to find derivatives in calculus.
In conclusion, the product rule for derivatives provides a straightforward method for finding the derivative of a product of two functions. It involves differentiating each function separately and combining the results using the given formula. Understanding and applying the product rule is essential for mastering calculus and solving a wide range of mathematical problems.
